The Existence Of Solution To Biharmonic Equation Without(AR)Condition In R~N

In this paper,we study the following biharmonic problem:where N?5.We assume that the nonlinearity f satisfies the following conditions:(Hi)(i)f:RN×R?R is a Caratheodory function,with f(x,s)s>0,for every(x,s)?RN x R,and f(x,0)?0,(?)x?RN;(ii)f(x,s)is bounded with respect to x,and f(x,s)is 1-periodic in xi,1?i?N.(H2)There is p?(2,2N/N-4),if N?5,such that,uniformly in(H3)(?)f(x,s)/s=0,uniformly in x?RN.(H4)There is a a?(0,?],such that,uniformly in x?RN.Under the conditions(H1)-(H4),with different parameter a,we prove the existence of nontrivial solution of the given biharmonic problem.